New classes of nonassociative divison algebras and MRD codes - Thompson, Daniel (2021)
In the first part of the thesis, we generalize a construction by J Sheekey that employs skew polynomials to obtain new nonassociative division algebras and maximum rank distance (MRD) codes. This construction contains Albert’s twisted fields as special cases. As a byproduct, we obtain a class of nonassociative real division algebras of dimension four which has not been described in the literature so far in this form. We also obtain new MRD codes. In the second part of the thesis, we study a general doubling process (similar to the one that can be used to construct the complex numbers from pairs of real numbers) to obtain new non-unital nonassociative algebras, starting with cyclic algebras. We investigate the automorphism groups of these algebras and when they are division algebras. In particular, we obtain a generalization of Dickson’s commutative semifields. We are using methods from nonassociative algebra throughout.